Existence of solutions to a phase field model of dynamic fracture with a crack dependent dissipation

We propose a phase-field model of dynamic crack propagation based on the
Ambrosio-Tortorelli approximation, which takes in account dissipative
effects due to the speed of the crack tips. In particular, adapting the
time discretization scheme contained in [Bourdin et al., Int. J.
Fracture 168 (2011), 133-143] and [Larsen et al., Math. Models Methods
Appl. Sci. 20 (2010), 1021-1048], we show the existence of a dynamic
crack evolution satisfying an energy dissipation balance, according to
Griffith's criterion.